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On the approximation of free discontinuity problems. (English) Zbl 0776.49029
This paper compliments another one of the authors [Commun. Pure Appl. Math. 43, No. 8, 999-1036 (1990; Zbl 0722.49020)] both concerning the approximation (in the sense of \(\Gamma\)-convergence) of the Mumford-Shah type functional (or rather its lower semicontinuous envelope) by elliptic functionals which formally have simpler form.
The Mumford-Shah type functionals appear in a variational approach to the image segmentation problem and have as unknown also a hypersurface \(K\subset \mathbb{R}^ n\): \[ F(u,K)=\int_{\Omega\backslash K} (\alpha|\nabla u|^ 2 + \beta(u-g)^ 2)dx +{\mathcal H}^{n- 1}(K), \] where \(\Omega\) is an open and bounded set in \(\mathbb{R}^ n\), \(K\) is closed in \(\Omega\), \(u\in C^ 1(\Omega\backslash K)\), \(\alpha\), \(\beta\) are fixed positive constants and \(g\) is a given function in \(L^ \infty(\Omega)\), \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure. Each of the two papers proposes a variational approximation of the lower semicontinuous envelope \(\overline F\) by two different families of functionals \(F_ h(u,s)\), \(G_ x(u,s)\), resp., \(s\) being now a functional variable, so that the minimizers “converge” (up to subsequences) to a minimizer of \(\overline F\). The so-called minimal partition problem is also considered.

49R50 Variational methods for eigenvalues of operators (MSC2000)
49Q12 Sensitivity analysis for optimization problems on manifolds
68U10 Computing methodologies for image processing